From finite geometry exact quantities to (elliptic) scattering amplitudes for spin chains: the 1/2-XYZ

TL;DR

This paper derives nonlinear integral equations for the spin 1/2-XYZ chain in the disordered regime, connecting finite geometry models to scattering amplitudes in elliptic and sine-Gordon theories, and explicitly confirms two known S-matrices.

Contribution

It introduces a novel nonlinear integral equation framework for the XYZ chain, extending it to excitations and deriving explicit scattering matrices, linking finite spin chains to integrable quantum field theories.

Findings

Derived nonlinear integral equations for the XYZ chain in the disordered regime.

Connected the scaling limit to sine-Gordon theory on cylindrical geometry.

Explicitly confirmed two known S-matrices as scattering descriptions.

Abstract

Initially, we derive a nonlinear integral equation for the vacuum counting function of the spin 1/2-XYZ chain in the {\it disordered regime}, thus paralleling similar results by Kl\"umper \cite{KLU}, achieved through a different technique in the {\it antiferroelectric regime}. In terms of the counting function we obtain the usual physical quantities, like the energy and the transfer matrix (eigenvalues). Then, we introduce a double scaling limit which appears to describe the sine-Gordon theory on cylindrical geometry, so generalising famous results in the plane by Luther \cite{LUT} and Johnson et al. \cite{JKM}. Furthermore, after extending the nonlinear integral equation to excitations, we derive scattering amplitudes involving solitons/antisolitons first, and bound states later. The latter case comes out as manifestly related to the Deformed Virasoro Algebra of Shiraishi et al. \cite{SKAO}. Although this nonlinear integral equations framework was contrived to deal with finite geometries, we prove it to be effective for discovering or rediscovering S-matrices. As a particular example, we prove that this unique model furnishes explicitly two S-matrices, proposed respectively by Zamolodchikov \cite{ZAMe} and Lukyanov-Mussardo-Penati \cite{LUK, MP} as plausible scattering description of unknown integrable field theories.